Research article

Hamilton’s gradient estimate for fast diffusion equations under geometric flow

  • Received: 08 January 2019 Accepted: 08 May 2019 Published: 15 May 2019
  • MSC : 53C21, 53C44, 58J35

  • Suppose that \lt i \gt M \lt /i \gt is a complete noncompact Riemannian manifold of dimension \lt i \gt n \lt /i \gt . In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations $ \dfrac{\partial u}{\partial t} = \Delta u^m ,\qquad 1-\dfrac{4}{n+8} \lt m \lt 1 $ on $M\times (-\infty, 0]$ under the geometric flow.

    Citation: Ghodratallah Fasihi-Ramandi. Hamilton’s gradient estimate for fast diffusion equations under geometric flow[J]. AIMS Mathematics, 2019, 4(3): 497-505. doi: 10.3934/math.2019.3.497

    Related Papers:

  • Suppose that \lt i \gt M \lt /i \gt is a complete noncompact Riemannian manifold of dimension \lt i \gt n \lt /i \gt . In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations $ \dfrac{\partial u}{\partial t} = \Delta u^m ,\qquad 1-\dfrac{4}{n+8} \lt m \lt 1 $ on $M\times (-\infty, 0]$ under the geometric flow.


    加载中


    [1] M. Bailesteanu, X. D. Cao, A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal., 258 (2010), 3517–3542. doi: 10.1016/j.jfa.2009.12.003
    [2] L. R. Evangelista, E. K. Lenzi, Fractional diffusion equations and anomalous diffusion, Cambridge University Press, 2018.
    [3] S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008–1023. doi: 10.1016/j.jfa.2008.05.014
    [4] H. Li, H. Bai, G. Zhang, Hamiltons gradient estimates for fast diffusion equations under the Ricci flow, J. Math. Anal. Appl., 444 (2016), 1372–1379. doi: 10.1016/j.jmaa.2016.07.017
    [5] G. A. Mendes, M. S. Ribeiro, R. S. Mendes, et al. Nonlinear Kramers equation associated with nonextensive statistical mechanics, Phys. Rev. E, 91 (2015), 052106. doi: 10.1103/physreve.91.052106
    [6] P. Li, S.-T. Yau, On the parabolic kernel of the Schrdinger operator, Acta Math., 156 (1986), 153–201. doi: 10.1007/bf02399203
    [7] S. P. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pac. J. Math., 243 (2009), 165–180. doi: 10.2140/pjm.2009.243.165
    [8] J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pac. J. Math., 253 (2011), 489–510. doi: 10.2140/pjm.2011.253.489
    [9] C. Tsallis, D. J. Bukman, Anomalous diffusion in the presence of external forces: Exact timedependent solutions and their thermostatistical basis, Phys. Rev. E, 54 (1996).
    [10] J. L. Vzquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Ser. Math. Appl., Vol. 33, Oxford University Press, Oxford, 2006.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2964) PDF downloads(604) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog